Five easy equations for patient flow through an emergency department

INTRODUCTION: Queue models are effective tools for framing management decisions and Danish hospitals could benefit from awareness of such models. Currently, as emergency departments (ED) are under reorganization, we deem it timely to empirically investigate the applicability of the standard “M/M/1” queue model in order to document its relevance. MATERIAL AND METHODS: We compared actual versus theoretical distributions of hourly patient flow from 27,000 patient cases seen at Frederiksberg Hospital’s ED. Formulating equations for arrivals and capacity, we wrote and tested a five equation simulation model. RESULTS: The Poisson distribution fitted arrivals with an hour-of-the-day specific parameter. Treatment times exceeding 15 minutes were well-described by an exponential distribution. The ED can be modelled as a black box with an hourly capacity that can be estimated either as admissions per hour when the ED operates full hilt Poisson distribution or from the linear dependency of waiting times on queue number. The results show that our ED capacity is surprisingly constant despite variations in staffing. These findings led to the formulation of a model giving a compact framework for assessing the behaviour of the ED under different assumptions about opening hours, capacity and workload. CONCLUSION: The M/M/1 almost perfectly fits our ED. Thus modeling and simulations have contributed to the management process. FUNDING: not relevant. TRIAL REGISTRATION: not relevant.